On maxima of Takagi-van der Waerden functions
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- by Yoshikazu Baba PDF
- Proc. Amer. Math. Soc. 91 (1984), 373-376 Request permission
Abstract:
Generalizing Takagi’s function ${F_2}\left ( x \right )$ and van der Waerden’s function ${F_{10}}\left ( x \right )$, we introduce a class of nowhere differentiable continuous functions ${F_r}\left ( x \right )$, $r \geqslant 2$. Some properties of ${F_r}\left ( x \right )$ concerning especially maxima are discussed. When $r$ is even, the Hausdorff dimension of the set of ${x^,}$’s giving the maxima of ${F_r}\left ( x \right )$ is proved to be $1/2$.References
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T. Takagi, A simple example of the continuous function without derivative, Proc. Phys.-Math. Soc. Tokyo Ser. II 1 (1903), 176-177.
- B. L. van der Waerden, Ein einfaches Beispiel einer nicht-differenzierbaren stetigen Funktion, Math. Z. 32 (1930), no. 1, 474–475 (German). MR 1545179, DOI 10.1007/BF01194647 B. Martynov, On maxima of the van der Waerden function, Kvant, June 1982, 8-14. (Russian)
- Masaya Yamaguti and Masayoshi Hata, Weierstrass’s function and chaos, Hokkaido Math. J. 12 (1983), no. 3, 333–342. MR 719972, DOI 10.14492/hokmj/1470081010
Additional Information
- © Copyright 1984 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 91 (1984), 373-376
- MSC: Primary 26A27
- DOI: https://doi.org/10.1090/S0002-9939-1984-0744632-0
- MathSciNet review: 744632